Post-Newtonian Approximationbcc.impan.pl/13Gravitational/uploads/lecture2.pdf · Post-Newtonian gravity and gravitational-wave astronomy Polarization waveforms in the SSB reference

Post-Newtonian gravity and gravitational-wave astronomyPolarization waveforms in the SSB reference frame

Relativistic binary systemsEffective one-body formalism

Post-Newtonian Approximation

Piotr Jaranowski

Faculty of Physcis, University of Bia lystok, Poland

01.07.2013

P. Jaranowski School of Gravitational Waves, 01–05.07.2013, Warsaw



1 Post-Newtonian gravity and gravitational-wave astronomy

2 Polarization waveforms in the SSB reference frame

3 Relativistic binary systemsLeading-order waveforms (Newtonian binary dynamics)Leading-order waveforms without radiation-reaction effectsLeading-order waveforms with radiation-reaction effectsPost-Newtonian correctionsPost-Newtonian spin-dependent effects

4 Effective one-body formalismEOB-improved 3PN-accurate HamiltonianUsage of Pade approximantsEOB flexibility parameters











We will concentrate in this lecture to the most important applicationof post-Newtonian approximation to general relativity:relativistic two-body problem,i.e. the problem of finding motion and gravitational radiationof selfgravitating relativistic systemswhich consist of two extended bodies.

Relativistic binary systems exist in nature,they comprise compact objects:neutron stars or black holes.These systems emit gravitational waves,which experimenters try to detectwithin the LIGO/VIRGO/GEO600 projects.




We will concentrate in this lecture to the most important applicationof post-Newtonian approximation to general relativity:relativistic two-body problem,i.e. the problem of finding motion and gravitational radiationof selfgravitating relativistic systemswhich consist of two extended bodies.

Relativistic binary systems exist in nature,they comprise compact objects:neutron stars or black holes.These systems emit gravitational waves,which experimenters try to detectwithin the LIGO/VIRGO/GEO600 projects.




Black-hole binary: Inspiral—merger—ringdown

As the result of emission of gravitational waves,the size of the binary orbit decreasesand the binary components move faster,consequently leading to emission of gravitational waveswith increasing amplitude and frequency—producing a gravitational-wave chirp signal.

-15 -10 -5 0 5 10 15

-15

-10

-5

0

5

10

15

x

y




To detect in noise of a detector weak gravitational waves(coming e.g. from a coalescing compact binary system)by means of matched filtering method,one has to construct a discrete bank of templates (waveforms)parametrized by possible values of the GW signal’s parameters(e.g. masses and spins of binary components).To ensure the succesful detection, bank of templateshas to cover the signal’s parameter space densly enough.

For construction of templates (waveforms) one has to knowtime evolution of the source of gravitational waves(i.e. compact binary system in the rest of this lecture).




To detect in noise of a detector weak gravitational waves(coming e.g. from a coalescing compact binary system)by means of matched filtering method,one has to construct a discrete bank of templates (waveforms)parametrized by possible values of the GW signal’s parameters(e.g. masses and spins of binary components).To ensure the succesful detection, bank of templateshas to cover the signal’s parameter space densly enough.

For construction of templates (waveforms) one has to knowtime evolution of the source of gravitational waves(i.e. compact binary system in the rest of this lecture).




Different approaches for solving relativistic two-body problem:

numerical relativity (breakthrough in 2005);

approximate ‘analytical’ methods:

post-Newtonian expansion ,

perturbation-based self-force approach.

(Blanchet et al., Phys. Rev. D 81, 064004 (2010))

PN expansion:0th order—Newtonian gravity;nPN order—corrections of order(v

c

)2n

∼(Gm

rc2

)n

to the Newtonian gravity.

Perturbation approach:m1

m2� 1.




Different approaches for solving relativistic two-body problem:

numerical relativity (breakthrough in 2005);

approximate ‘analytical’ methods:

post-Newtonian expansion ,

perturbation-based self-force approach.

(Blanchet et al., Phys. Rev. D 81, 064004 (2010))

PN expansion:0th order—Newtonian gravity;nPN order—corrections of order(v

c

)2n

∼(Gm

rc2

)n

to the Newtonian gravity.

Perturbation approach:m1

m2� 1.




Construction of bank of templates requires multiple integration of

partial differential equations(numerical relativity),

ordinary differential equations(approximate ‘analytical’ methods).

For GW signals coming from coalescence of (especially) spinning binaryblack holes (BBHs) with arbitrary mass ratios,due to limitations in available computing power,it will not be possible in the nearest futureto construct bank of templates based purely on numerical results.

An accurate equal-mass, non-spinning 8 orbit BBH simulationtakes ∼200 000 CPU hours.

The computational cost of a BBH simulation scales with themass ratio: an accurate 1:10 mass ratio, non-spinning 8 orbitsimulation would thus take ∼2 000 000 CPU hours.

(I.Hinder, Class. Quantum Grav. 27, 114004 (2010))
























Gravitational-wave induced evolution of BBHcomputed numerically and by means of the PN approximationsagree very well in the region, where the coalescing objectsare sufficiently far away.

Detection of GW signals (and extraction their parameters)from coalescing BBH by advanced LIGO/VIRGO detectors—the most promising waveforms are hybrid waveforms:PN (early inspiral) + numerical (late inspiral, merger, ringdown)or waveforms based on effective one-body formalism.




Gravitational-wave induced evolution of BBHcomputed numerically and by means of the PN approximationsagree very well in the region, where the coalescing objectsare sufficiently far away.

Detection of GW signals (and extraction their parameters)from coalescing BBH by advanced LIGO/VIRGO detectors—the most promising waveforms are hybrid waveforms:PN (early inspiral) + numerical (late inspiral, merger, ringdown)or waveforms based on effective one-body formalism.




Effective one-body (EOB) formalism is a formalism,based on approximate PN and BH perturbation theory results,which allows to model ‘analytically’ the whole coalescenceof a BBH system, from its adiabatic inspiral up to vibrationsof the ‘resultant’ Kerr BH.It is being developed by Damour and his collaborators since 1999.

Main idea of EOB approach: To extend the domain of validity of PNand BH perturbation theories so as to approximately covernon-perturbative features of BBH evolution.




Utility (partially expected) of EOB formalism

It provides accurate templates needed for detection of GW signals(and estimation of their parameters) originating from coalescencesof BBHs,

it gives a physical understanding of dynamics and radiation of BBHs,

it can be extended to BH-NS or NS-NS systems(after incorporating tidal interactions).




Black-hole binary: EOB vs numerical relativity

T.Damour, A.Nagar, Phys.Rev.D 79, 081503(R) (2009)

500 1000 1500 2000 2500 3000 3500 4000

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

t

ℜ[Ψ

22]/ν

Numerical Relativity (Caltech-Cornell)EOB (a5 = 0; a6=-20)

1:1 mass ratio

3800 3820 3840 3860 3880 3900 3920 3940 3960 3980 4000

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

t

Merger time











In the coordinate system (t, x is ) centered on a gravitational-wave source

the wave polarization functions can be computed from the equations

h+(t, x is ) =

G

c4 R

(d2Jxx

dt2

(t − R

c

)− d2Jyy

dt2

(t − R

c

)), (1a)

h×(t, x is ) =

2G

c4 R

d2Jxy

dt2

(t − R

c

), (1b)

for the wave propagating in the +z direction of the coordinate system.




Let us introduce coordinate system (t, x i ) about which we assume thatsome solar-system related observer measures the gravitational-wave fieldhTTµν in the neighborhood of its spatial origin.

In the discussion of the detection of gravitational waves bysolar-system-based detectors, the origin of these coordinateswill be located at the solar system barycenter (SSB).

Let us denote by x∗ the constant 3-vector joiningthe origin of the coordinates (x i ) (i.e. the SSB)with the origin of the (x i

s ) coordinates (i.e. the ‘center’ of the source).

We also assume that the spatial axes in both coordinate systems areparallel to each other, i.e. the coordinates x i and x i

s differ just by constantshifts determined by the components of the 3-vector x∗,

x i = x∗i + x is . (2)

It means that in both coordinate systems the components hTTµν of the

gravitational-wave field are numerically the same.




Equations (1) are valid only when the z axis of the TT coordinate systemis parallel to the 3-vector x∗ − x joining the observer located at x(x is the 3-vector joining the SSB with the observer’s location)and the gravitational-wave source at the position x∗.

If one changes the location x of the observer, one has to rotate the spatialaxes to ensure that Eqs. (1) are still valid.

Let us now fix, in the whole region of interest, the direction of the +z axisof both coordinate systems considered here,by choosing it to be antiparallel to the 3-vector x∗ (so for the observerlocated at the SSB gravitational wave propagates along the +z direction).

To a very good accuracy one can assume that the size of the region wherethe observer can be located is very small compared to the distance fromthe SSB to the gravitational-wave source, which is equal to

r∗ := |x∗|.

Our assumption thus means that r � r∗, where r := |x|.




Then x∗ = (0, 0,−r∗) and the Taylor expansion of R = |x− x∗| andn := (x− x∗)/R around x∗ reads

|x− x∗| = r∗(

1 +z

r∗+

x2 + y 2

2(r∗)2+O

((x l/r∗)3)), (3a)

n =

(− x

r∗+

xz

2(r∗)2,− y

r∗+

yz

2(r∗)2, 1− x2 + y 2

2(r∗)2

)+O

((x l/r∗)3).

(3b)

One can compute the TT projection of the reduced mass quadrupolemoment at the point x in the direction of the unit vector n (which ingeneral is not parallel to the +z axis). One gets

J TTxx = −J TT

yy =1

2(Jxx − Jyy ) +O

(x l/r∗

), (4a)

J TTxy = J TT

yx = Jxy +O(x l/r∗

), (4b)

J TTzi = J TT

iz = 0 +O(x l/r∗

)for i = x , y , z . (4c)




To obtain the gravitational-wave field hTTµν in the coordinate system (t, x i )

one should plug Eqs. (4) into the quadrupole formula.

If one neglects in Eqs. (4) the terms of the order of x l/r∗, then in thewhole region of interest covered by the single TT coordinate system,the gravitational-wave field hTT

µν can be written in the form

hTTµν (t, x) = h+(t, x) e+

µν + h×(t, x) e×µν +O(x l/r∗

), (5)

where e+ and e× are the polarization tensors and the functions h+ andh× are of the form given in Eqs. (1).




Dependence of the 1/R factor in the amplitude of the wavepolarizations (1) on the observer’s position x (with respect to the SSB)is usually negligible, so 1/R in the amplitudes can be replaced by 1/r∗

[this is consistent with the neglection of the O(x l/r∗) termswe have just made in Eqs. (4)].

This is not the case for the 2nd time derivative of Jij in (1),which determines the time evolution of the wave polarizations phasesand which is evaluated at the retarded time t − R/c.Here it is usually enough to take into account the first correction to r∗

[given by Eq. (3)].




After taking all this into account the wave polarizations (1) take the form

h+(t, x i ) =G

c4 r∗

(d2Jxx

dt2

(t − z + r∗

c

)− d2Jyy

dt2

(t − z + r∗

c

)), (6a)

h×(t, x i ) =2G

c4 r∗d2Jxy

dt2

(t − z + r∗

c

). (6b)

The wave polarization functions (6) represent a plane gravitational wavepropagating in the +z direction.




Leading-order waveforms (Newtonian binary dynamics)Leading-order waveforms without radiation-reaction effectsLeading-order waveforms with radiation-reaction effectsPost-Newtonian correctionsPost-Newtonian spin-dependent effects

















We consider a binary system made of two bodies with masses m1 and m2

(we will always assume m1 ≥ m2). We introduce

M := m1 + m2, µ :=m1m2

M, (7)

so M is the total mass of the system and µ is its reduced mass.

It is also useful to introduce the dimensionless symmetric mass ratio

ν :=m1m2

M2=

µ

M. (8)

The quantity ν satisfies 0 ≤ ν ≤ 1/4, the case ν = 0 corresponds to thetest-mass limit and ν = 1/4 describes equal-mass binary.

We start from deriving the wave polarization functions h+ and h×for waves emitted by a binary system in the casewhen the dynamics of the binary can reasonably be describedwithin the Newtonian theory of gravitation.





Let r1 and r2 denote the position vectors of the bodies, i.e. the 3-vectorsconnecting the origin of some reference frame with the bodies.We introduce the relative position vector,

r12 := r1 − r2. (9)

The center-of-mass reference frame is defined by the requirement that

m1 r1 + m2 r2 = 0. (10)

Solving Eqs. (9) and (10) with respect to r1 and r2 one gets

r1 =m2

Mr12, r2 = −m1

Mr12. (11)





In the center-of-mass reference frame we introduce the spatial coordinates(xc, yc, zc) such that the total orbital angular momentum vector J of thebinary is directed along the +zc axis.

Then the trajectories of both bodies lie in the (xc, yc) plane, so theposition vector ra of the ath body (a = 1, 2) has components

ra = (xca, yca, 0),

and the relative position vector components are

r12 = (xc12, yc12, 0),

where xc12 := xc1 − xc2 and yc12 := yc1 − yc2.

It is convenient to introduce in the coordinate (xc12, yc12) planethe usual polar coordinates (r , φ):

xc12 = r cosφ, yc12 = r sinφ. (12)





Within the Newtonian gravity the orbit of the relative motion is an ellipse(we consider here only gravitationally bound binaries).We place the focus of the ellipse at the origin of the (xc12, yc12)coordinates. In polar coordinates the ellipse is described by the equation

r(φ) =a(1− e2)

1 + e cos(φ− φ0), (13)

where a is the semimajor axis, e is the eccentricity,and φ0 is the angle of the orbital periapsis.

The time dependence of the relative motion is determinedby the Kepler’s equation,

φ =2π

P(1− e2)−3/2(1 + e cos(φ− φ0)

)2, (14)

where P is the orbital period of the binary,

P = 2π

√a3

GM. (15)





The binary’s binding energy E and the modulus J of its total angularorbital momentum are related to the parameters of the relative orbitthrough the equations

E = −GMµ

2a, J = µ

√GMa(1− e2). (16)





Let us now introduce the TT ‘wave’ coordinates (xw, yw, zw)in which the gravitational wave is traveling in the +zw directionand with the origin at the SSB.

The line along which the plane tangent to the celestial sphere at thelocation of the binary’s center-of-mass[this plane is parallel to the (xw, yw) plane]intersects the orbital (xc, yc) plane is called the line of nodes.

Let us adjust the center-of-mass and the wave coordinatesin such a way, that the x axes of both coordinate systemsare parallel to each other and to the line of nodes.





Then the relation between these coordinates is determined by the rotationmatrix S,xw

ywzw

=

x∗wy∗wz∗w

+ S

xcyczc

, S :=

1 0 00 cos ι sin ι0 − sin ι cos ι

, (17)

where (x∗w, y∗w, z

∗w) are the components of the vector x∗ joining the SSB

and the binary’s center-of-mass, and ι ( 0 ≤ ι ≤ π) is the angle betweenthe orbital angular momentum vector J of the binary and the line of sight(i.e. the +zw axis).

We assume that the center of mass of the binaryis at rest with respect to the SSB.





In the center-of-mass reference frame the binary’s moment of inertiatensor has changing in time components which are equal

I ijc (t) = m1 xic1(t) x j

c1(t) + m2 xic2(t) x j

c2(t). (18)

Making use of Eqs. (11) one finds the matrix Ic built from the I ijccomponents of the inertia tensor,

Ic(t) = µ

(xc12(t))2 xc12(t) yc12(t) 0xc12(t) yc12(t) (yc12(t))2 0

0 0 0

. (19)





The inertia tensor components in wave coordinates are related with thosein center-of-mass coordinates through the relation

I ijw(t) =3∑

k=1

3∑`=1

∂x iw

∂xkc

∂x jw

∂x`cIk`c (t), (20)

what in matrix notation reads

Iw(t) = S · Ic(t) · ST. (21)





To obtain the wave polarization functions h+ and h× we plug thecomponents of the binary’s inertia tensor into general equations[note that in these equations the components Jij of the reducedquadrupole moment can be replaced by the components Iijof the inertia tensor].

We get [here R = |x∗| is the distance to the binary’s center-of-mass andtr = t − (zw + R)/c is the retarded time]

h+(t, x) =Gµ

c4R

(sin2 ι

(r(tr)

2 + r(tr)r(tr))

+ (1 + cos2 ι)(r(tr)

2 + r(tr)r(tr)− 2r(tr)2φ(tr)

2) cos 2φ(tr)

− (1 + cos2 ι)(4r(tr)r(tr)φ(tr) + r(tr)

2φ(tr))

sin 2φ(tr)),

(22a)

h×(t, x) =2Gµ

c4Rcos ι

((4r(tr)r(tr)φ(tr) + r(tr)

2φ(tr))

cos 2φ(tr)

+(r(tr)

2 + r(tr)r(tr)− 2r(tr)2φ(tr)

2) sin 2φ(tr)). (22b)













Gravitational waves emitted by the binary diminish the binary’sbinding energy and total orbital angular momentum.This makes the orbital parameters changing in time.

Let us first assume that these changes are so slow that they can beneglected during the time interval in which the observations areperformed.

Making use of Eqs. (14) and (13) one can then eliminate from theformulae (22) the first and the second time derivatives of r and φ,treating the parameters of the orbit, a and e, as constants.

r(φ) =a(1− e2)

1 + e cos(φ− φ0)(13)

φ =2π

P(1− e2)−3/2(1 + e cos(φ− φ0)

)2(14)





The result is the following:

h+(t, x) =4G 2Mµ

c4Ra(1− e2)

(A0(tr) + A1(tr) e + A2(tr) e

2), (23a)

h×(t, x) =4G 2Mµ

c4Ra(1− e2)cos ι

(B0(tr) + B1(tr) e + B2(tr) e

2), (23b)

where the functions Ai and Bi , i = 0, 1, 2, are defined as follows

A0(t) = −1

2(1 + cos2

ι) cos 2φ(t),

A1(t) =1

4sin2

ι cos(φ(t)− φ0)−1

8(1 + cos2

ι)(

5 cos(φ(t) + φ0) + cos(3φ(t)− φ0)),

A2(t) =1

4sin2

ι−1

4(1 + cos2

ι) cos 2φ0,

B0(t) = − sin 2φ(t),

B1(t) = −1

4

(sin(3φ(t)− φ0) + 5 sin(φ(t) + φ0)

),

B2(t) = −1

2sin 2φ0.





In the case of circular orbits the eccentricity e vanishesand the wave polarization functions (23) simplify to

h+(t, x) = −2G 2Mµ

c4Ra(1 + cos2 ι) cos 2φ(tr), (25a)

h×(t, x) = −4G 2Mµ

c4Racos ι sin 2φ(tr), (25b)

where a is the radius of the relative circular orbit and

φ(t) = φ0 +2π

P(t − t0).













Let us now consider observational intervals short enoughso it is necessary to take into account effects of radiation reactionchanging the parameters of the bodies’ orbits.

In the leading order the rates of emission of energy and angularmomentum carried by gravitational waves are given by the formulae

LgwE =

G

5c5

3∑i=1

3∑j=1

⟨(d3Jij

dt3

)2⟩, Lgw

Ji=

2G

5c5

3∑j=1

3∑k=1

3∑`=1

εijk

⟨d2Jj`

dt2

d3Jk`

dt3

⟩.

We plug into them the Newtonian trajectories of the bodies and performtime averaging over one orbital period of the motion. We get

LgwE =

32

5

G 4µ2M3

c5a5(1− e2)7/2

(1 +

73

24e2 +

37

96e4), (26a)

LgwJ =

32

5

G 7/2µ2M5/2

c5a7/2(1− e2)2

(1 +

7

8e2). (26b)





It is easy to obtain equations describing the leading-order timeevolution of the relative orbit parameters a and e.

We first differentiate both sides of equations

E = −GMµ

2a, J = µ

√GMa(1− e2),

with respect to time treating a and e as functions of time.Then the rates of changing the binary’s binding energy and angularmomentum, dE/dt and dJ/dt, one replaces by minus the rates ofemission of respectively energy Lgw

E and angular momentum LgwJ carried

by gravitational waves, given in Eqs. (26).





The two equations such obtained can be solvedwith respect to the derivatives da/dt and de/dt:

da

dt= −64

5

G 3µM2

c5a3(1− e2)7/2

(1 +

73

24e2 +

37

96e4), (27a)

de

dt= −304

15

G 3µ2M2

c5a4(1− e2)5/2e(

1 +121

304e2). (27b)

Let us note two interesting features of the evolutiondescribed by the above equations:

Eq. (27b) implies that initially circular orbits (for which e = 0) remaincircular during the evolution;

the eccentricity e of the relative orbit decreases with time much morerapidly than the semimajor axis a, so gravitational-wave reaction inducesthe circularization of the binary orbits(roughly, when the semimajor axis is halved,the eccentricity goes down by a factor of three).





Evolution of the orbital parameters caused by the radiation reactioninfluences gravitational waveforms. To see this influence we restrict ourconsiderations to binaries along circular orbits.

The decay of the radius a of the relative orbit is given by the equation[this is Eq. (27a) with e = 0]

da

dt= −64

5

G 3µM2

c5a3. (28)

Integration of this equation leads to

a(t) = a0

(tc − t

tc − t0

)1/4

, (29)

where a0 := a(t = t0) is the initial value of the orbital radius and tc − t0 isthe formal ‘lifetime’ or ‘coalescence time’ of the binary, i.e. the time afterwhich the distance between the bodies is zero.





It means that tc is the moment of the coalescence, a(tc) = 0; it is equal to

tc = t0 +5

256

c5a40

G 3µM2. (30)

If one neglects radiation reaction effects, then along the Newtoniancircular orbit the time derivative of the orbital phase, i.e. the orbitalangular frequency ω, is constant and given by [see Eqs. (14) and (15)]

ω = φ =2π

P=

√GM

a3. (31)

We now assume that the inspiral of the binary system caused by theradiation reaction is adiabatic, i.e. it can be thought as a sequence ofcircular orbits with radii a(t) which slowly diminish in timeaccording to Eq. (29).

Adiabacity thus means that Eq. (31) is valid during the inspiral,but now angular frequency ω is a function of time,because of time dependence of the orbital radius a.





Making use of Eqs. (29)–(31) we get the following equationfor the time evolution of the instantaneous orbital angular frequencyduring the adiabatic inspiral:

ω(t) = ω0

(1− 256(GM)5/3

5c5ω

8/30 (t − t0)

)−3/8

, (32)

where ω0 := ω(t = t0) is the initial value of the orbital angular frequencyand where we have introduced the new parameter M(with dimension of a mass) called a chirp mass of the system:

M := µ3/5M2/5. (33)





To get the polarization waveforms h+ and h× which describesgravitational radiation emitted during the adiabatic inspiral of the binarysystem, we use the general formulae (22).

In these formulae we neglect all the terms proportional to r ≡ a or r ≡ a(the radial coordinate r is identical with the radius a of the relativecircular orbit, r ≡ a).

By virtue of Eq. (31) φ ∝ a, therefore we also neglect terms ∝ φ.

All these simplifications lead to the following formulae:

h+(t, x) = −4(GM)5/3

c4R

1 + cos2 ι

2ω(tr)

2/3 cos 2φ(tr), (34a)

h×(t, x) = −4(GM)5/3

c4Rcos ι ω(tr)

2/3 sin 2φ(tr). (34b)





Making use of Eq. (31) one can express the time dependence of thewaveforms (34) in terms of the single function a(t):

h+(t, x) = −4G 2µM

c4R

1 + cos2 ι

2

1

a(tr)cos 2

(√GM

∫ tr

t0

a(t′)−3/2dt′ + φ0

),

(35a)

h×(t, x) = −4G 2µM

c4Rcos ι

1

a(tr)sin 2

(√GM

∫ tr

t0

a(t′)−3/2dt′ + φ0

),

(35b)

where φ0 := φ(t = t0) is the initial value of the orbital phase of the binary.





It is sometimes useful to analyze the chirp waveforms h+ and h× in termsof the instantaneous frequency fgw of the gravitational waves emitted by acoalescing binary system.

The frequency fgw is defined as

fgw(t) := 2× 1

2π

dφ(t)

dt, (36)

where the extra factor 2 is due to the fact that [see Eqs. (34)]the instantaneous phase of the gravitational waveforms h+ and h×is 2 times the instantaneous phase φ(t) of the orbital motion(so the gravitational-wave frequency fgw is twice the orbital frequency).





Making use of Eqs. (29)–(31) one gets that for the binary with circularorbits the gravitational-wave frequency fgw reads

fgw(t) =53/8c15/8

8πG 5/8

1

M5/8(tc − t)−3/8. (37)

The chirp waveforms given in Eqs. (35) can be rewritten in terms of thegravitational-wave frequency fgw as follows

h+(t, x) = −h0(tr)1 + cos2 ι

2cos(

2π

∫ tr

t0

fgw(t′)dt′ + 2φ0

), (38a)

h×(t, x) = −h0(tr) cos ι sin(

2π

∫ tr

t0

fgw(t′)dt′ + 2φ0

), (38b)

where h0(t) is the changing in time amplitude of the waveforms,

h0(t) :=4π2/3G 5/3

c4

M5/3

Rfgw(t)2/3. (39)













There are two problems, usually analyzed separately:

problem of finding equations of motion (EOM),

problem of computing gravitational-wave luminosities(of energy, angular momentum, and linear momentum).

PN corrections for EOM/luminosities without spin-dependent effects

EOM N 1PN 2PN 2.5PN 3PN 3.5PN 4PN 4.5PN 5PN 5.5PN 6PN 6.5PN

Energy/angularmomentum

luminosity — — — N — 1PN 1.5PN 2PN 2.5PN 3PN 3.5PN 4PN





There are two problems, usually analyzed separately:

problem of finding equations of motion (EOM),

problem of computing gravitational-wave luminosities(of energy, angular momentum, and linear momentum).

PN corrections for EOM/luminosities without spin-dependent effects

EOM N 1PN 2PN 2.5PN 3PN 3.5PN 4PN 4.5PN 5PN 5.5PN 6PN 6.5PN

Energy/angularmomentum

luminosity — — — N — 1PN 1.5PN 2PN 2.5PN 3PN 3.5PN 4PN





The response function of the laser-interferometric detector to gravitationalwaves from coalescing compact binary(made of nonspinning bodies) in circular orbits:

h(t) =C

D

(φ(t)

)2/3sin(2φ(t) + α

),

D is the distance of the binary to the Earth,C and α are some constants,and φ(t) is the orbital phase of the binary(so φ(t) ≡ dφ(t)/dt is the angular frequency).

Dimensionless post-Newtonian parameter for circular orbits:

x ≡ 1

c2(GMφ)2/3.





The orbital phase φ(t) evolution is computedfrom the balance equation

dE

dt= −L,

which has the following PN expansion:

d

dt

(EN +

1

c2E1PN +

1

c4E2PN +

1

c6E3PN +

1

c8E4PN +O

((v/c)9))

= −(LN +

1

c2L1PN +

1

c3L1.5PN +

1

c4L2PN +

1

c5L2.5PN

+1

c6L3PN +

1

c7L3.5PN +

1

c8L4PN +O

((v/c)9)).





Bounding energy in the center-of-mass frame for circular orbitsup to the 4PN order

E(x ; ν) = −µc2x

2

(1 + e1PN(ν) x + e2PN(ν) x2 + e3PN(ν) x3

+(e4PN(ν) +

448

15ν ln x

)x4 +O

((v/c)10)),

e1PN(ν) = −3

4−

1

12ν, e2PN(ν) = −

27

8+

19

8ν −

1

24ν

2,

e3PN(ν) = −675

64+

( 34445

576−

205

96π

2)ν −

155

96ν

2 −35

5184ν

3,

e4PN(ν) = −3969

128+ c1ν +

(−

498449

3456+

3157

576π

2)ν

2 +301

1728ν

3 +77

31104ν

4

(Jaranowski & Schafer 2012–2013, Foffa & Sturani 2013),

c1 = −123671

5760+

9037

1536π

2 +896

15(2 ln 2 + γ)) (γ is the Euler’s constant)

(Le Tiec, Blanchet, & Whiting 2012, Bini & Damour 2013).





Gravitational-wave luminosity for circular orbits up to the 3.5PN order

L(x ; ν) =32c5

5Gν2x5

(1 + `1PN(ν) x + 4π x3/2 + `2PN(ν) x2

+ `2.5PN(ν) x5/2 +(`3PN(ν)− 856

105ln(16x)

)x3

+ `3.5PN(ν) x7/2 +O((v/c)8)),

`1PN(ν) = −1247

336−

35

12ν, `2PN(ν) = −

44711

9072+

9271

504ν +

65

18ν

2,

`2.5PN(ν) =

(−

8191

672−

535

24ν

)π,

`3PN(ν) =6643739519

69854400+

16

3π

2 −1712

105γ +

(−

134543

7776+

41

48π

2)ν −

94403

3024ν

2 −775

324ν

3,

`3.5PN(ν) =

(−

16285

504+

214745

1728ν +

193385

3024ν

2)π.





Post-Newtonian results

Gravitational-wave luminosities for energy and angular momentum in thecase of quasi-elliptical orbits were computed up to the 3PN order beyondthe leading-order formula.

PN effects modify not only the orbital phase evolution of the binary,but also the amplitudes of the wave polarizations h+ and h×:—for quasi-circular orbits the 3PN-accurate corrections were computed;—for quasi-elliptical orbits the 2PN-accurate corrections were computed.

It is not easy to obtain the wave polarizations h+ and h× as explicitfunctions of time taking into account all known higher-orderPN corrections. In the case of quasi-elliptical orbits one has to dealwith three different time scales: orbital period, periastron precession,and radiation reaction time scale.













The spin of a rotating body is of the order

S ∼ mavspin,

where m and a denote the mass and typical size of the body, respectively,and vspin represents the velocity of the body’s surface.

We are interested in compact bodies, so

a ∼ Gm

c2, and then S ∼ Gm2 vspin

c2.





Nomenclature on PN spin-dependent effects

Maximally rotating bodies: vspin ∼ c =⇒ S ∼ Gm2

c.

Spin-orbit (i.e. linear in S) effects in EOM:

1.5PN + 2.5PN + · · · ;

spin-spin effects in EOM:

2PN + 3PN + · · · .

Slowly rotating bodies: vspin � c =⇒ S ∼ Gm2vspin

c2.

Spin-orbit (i.e. linear in S) effects in EOM:

2PN + 3PN + · · · ;

spin-spin effects in EOM:

3PN + 3PN + · · · .





More nomenclature on PN spin-dependent effects

Just words, no 1/c counting:leading-order (LO) + next-to-leading-order (NLO) +next-to-next-to-leading-order (NNLO) + · · · .Formal counting:PN orders are counted in terms of 1/c originally present in the Einsteinequations, i.e. the spin variables do not contribute to counting of 1/c.Then, e.g., spin-orbit effects in EOM start as follows:

1PN + 2PN + · · · .




EOB-improved 3PN-accurate HamiltonianUsage of Pade approximantsEOB flexibility parameters









Structure of EOB formalism

1 Computation of inspiral + plunge waveform hinsplunge(t):

PN conservative dynamics −→ EOB-improved Hamiltonian;

PN radiation losses −→ EOB radiation-reaction force;PN waveform.

2 Computation of ringdown waveform hringdown(t):

BH perturbation theory.

EOB-waveform: hEOB(t) = θ(tm− t) hinsplunge(t) + θ(t− tm) hringdown(t),

where θ(t) denotes Heaviside’s step function and tm is the time at whichthe two waveforms hinsplunge, hringdown are matched.













Within the ADM approach the 3PN-accurate 2-point-massconservative Hamiltonian H

(x1, x2,p1,p2

)was derived.

We need its reduction to the center-of-mass frame.

Relative motion in the center-of-mass frame (p1 + p2 = 0):(x1, x2,p1,p2

)→ (q,p);

reduced variables in the center-of-mass frame:

q ≡ x1 − x2

GM, p ≡ p1

µ= −p2

µ, t ≡ t

GM.





‘Non-relativistic’ Hamiltonian of the 2-point-mass system:

HNR ≡ H − (m1 + m2)c2

µ.

Conservative 3PN-accurate Hamiltonian describing relative motion

HNR (q,p) = HN (q,p) +1

c2H1PN (q,p)

+1

c4H2PN (q,p) +

1

c6H3PN (q,p) ,

HN (q,p) =1

2p2 − 1

q, . . .

(the functional form of HNR (q,p) is gauge dependent).





Units: c = G = 1.

Real two-body problem(two masses m1, m2 orbiting around each other)

lEffective one-body problem

(one test particle of mass m0

moving in some background metric g effectiveαβ )





The mapping rules between the two problems(motivated by quantum considerations):

the adiabatic invariants (the action variables) Ii =∮pi dqi

are identified in the two problems;

the energies are mapped through a function f :

Eeffective = f (Ereal),

f is determined in the process of matching.

One looks for a metric g effectiveαβ such that the energies of the bound states

of a particle moving in g effectiveαβ are in one-to-one correspondence with the

energies of the two-body bound states:

Eeffective(Ii ) = f (Ereal(Ii )) .





Static and spherically symmetric effective metric(in Schwarzschild-like coordinates):

ds2eff = −A(q′)dt ′2 +

D(q′)

A(q′)dq′2 + q′2 (dθ′2 + sin2 θ′ dϕ′2),

A(q′) = 1 + a1M0

q′+ a2

(M0

q′

)2

+ a3

(M0

q′

)3

+ a4

(M0

q′

)4

+ · · · ,

D(q′) = 1 + d1M0

q′+ d2

(M0

q′

)2

+ d3

(M0

q′

)3

+ · · · .





The energy-map for the “non-relativistic” energies

ENReff ≡ ER

eff −m0 and ENRreal ≡ ER

real −M:

ENReff

m0=ENR

real

µ

(1 + α1

ENRreal

µ+ α2

(ENRreal

µ

)2

+ α3

(ENRreal

µ

)3

+ · · ·).

It is natural to require that

m0 ≡ µ = m1 m2/(m1 + m2),

M0 ≡ M = m1 + m2,

with these choices the Newtonian limit tells us that a1 = −2.





At each PN level one has only three arbitrary coefficients:the 1PN level: a2, d1, and α1;the 2PN level: a3, d2, and α2;the 3PN level: a4, d3, and α3.

How many independent equations has to be satisfiedwhen mapping the real problem onto the effective one?





The structure of the nPN conservative relative-motion realHamiltonian in the center-of-mass frame:

HNRnPN(q,p) = p2(n+1) +

1

q

(p2n + p2(n−1)(np)2 + · · ·+ (np)2n

)+

1

q2

(p2(n−1) + · · ·+ (np)2(n−1)

)+ · · ·+ 1

qn+1;

HNRnPN(q,p) has CH(n) =

(n + 1)(n + 2)

2+ 1 coefficients.





The identification of the action variables guaranteesthat the two problems are mapped by a canonical transformation,with generating function

G (q, p′) = qip′i + G (q, p′),

so the relation between the real phase-space coordinates (qi , pi )and the effective phase-space coordinates (q′i , p′i ) reads

q′i = qi +∂ G (q, p′)

∂ p′i, pi = p′i +

∂ G (q, p′)

∂ qi.





The structure of the nPN generating function:

GnPN(q,p′) = (q · p′){p′2n +

1

q

(p′2(n−1) + · · ·+ (np′)2(n−1)

)+ · · ·+ 1

qn

};

GnPN(q,p′) has CG (n) =n(n + 1)

2+ 1 coefficients.





The difference ∆(n) between the number of equations to satisfyand the number of unknowns at the nPN level:

∆(n) = CH(n)− (CG (n) + 3) = n − 2 ;

∆(1) = −1 ,

∆(2) = 0 ,

∆(3) = +1 .





2PN-accurate effective (geodesic) Hamiltonian

0 = µ2 + gαβeff (x ′) p′α p′β

⇓

HReff(q′,p′) =

√√√√A(q′)

(1 + p′2 +

(A(q′)

D(q′)− 1

)(n′ · p′)2

).





Matching results at the 1PN level

One unrestricted degree of freedom is used to impose the conditiond1 = 0, i.e. that the linearized effective metric coincides with thelinearized Schwarzschild metric.

Energy map coefficient: α1 =1

2ν.


No degrees of freedom left.

Energy map coefficient: α2 = 0.






One unrestricted degree of freedom is used to impose the conditiond1 = 0, i.e. that the linearized effective metric coincides with thelinearized Schwarzschild metric.

Energy map coefficient: α1 =1

2ν.


No degrees of freedom left.

Energy map coefficient: α2 = 0.





3PN-accurate effective (nongeodesic) Hamiltonian

0 = µ2 + gαβeff (x ′) p′α p′β + Aαβγδ(x ′) p′α p

′β p′γ p′δ

⇓

HReff(q′,p′) =

√√√√A(q′)

(1 + p′2 +

(A(q′)

D(q′)− 1

)(n′ · p′)2 + Z (q′,p′)

),

where Z (q′,p′) ≡ 1

q′2

(z1(p′2)2 + z2 p

′2(n′ · p′)2 + z3(n′ · p′)4).






The parameters z1, z2, and z3 satisfy the linear constraint:

8z1 + 4z2 + 3z3 = 6(4− 3ν)ν .

Energy map coefficient:α3 = 0 .

To simplify the 3PN effective dynamics of circular orbits, one chooses

z1 = 0 = z2 , z3 = 2(4− 3ν)ν .





The values α1 = 12ν at 1PN, α2 = 0 at 2PN, and α3 = 0 at 3PN

correspond exactly to

EReff

µ=

(ERreal)

2 −m21 −m2

2

2m1m2.

Solving above equation with respect to ERreal one obtains

the real EOB-improved 3PN-accurate Hamiltonian:

HRreal(q,p) = M

√1 + 2ν

HReff

(q′(q,p),p′(q,p)

)− µ

µ.













Pade approximant of (k, l)-type (with k + l = n) for seriesσ(x) = c0 + c1 x + · · ·+ cn x

n (with c0 6= 0):

Pkl (σ(x)) ≡ Nk(x)

Dl(x),

where the polynomials Nk (of degree k) and Dl (of degree l)are such that the Taylor expansion of Pk

l (σ(x)) coincideswith σ(x) up to O(xn) terms.





Let us consider the reduced angular momentum of the systemin the center-of-mass reference frame:

j ≡ JGm1m2

.

In the test-mass limit and along circular orbits:

j(x ; ν = 0) =1√

x(1− 3x),

the pole x = 1/3 corresponds to ‘light ring’(or the last unstable circular orbit).





As a result of the 3PN-accurate PN computations one gets

j2(x ; ν) =1

x

[1 +

1

3(9 + ν) x +

1

36(36− ν)(9− 4ν) x2 +

16

3w5(ν) x3

],

where

w5(ν) ≡ 81

16+

1

64

(41π2 − 7321

6

)ν +

23

64ν2 +

1

216ν3.

One can construct the following sequence of near-diagonal Pades of j2(x):

j2P1

(x ; ν) ≡ 1

xP0

1

[1 +

1

3(9 + ν) x

],

j2P2

(x ; ν) ≡ 1

xP1

1

[1 +

1

3(9 + ν) x +

1

36(36− ν)(9− 4ν) x2

],

j2P3

(x ; ν) ≡ 1

xP2

1

[1 +

1

3(9 + ν) x +

1

36(36− ν)(9− 4ν) x2 +

16

3w5(ν) x3

].





The results of computation read:

j2P1

(x ; ν) =1

x(1−

(3 + 1

3ν)x) ,

j2P2

(x ; ν) =1 + 1

9ν + 25

12νx

x(1 + 1

9ν −

(3− 17

12ν + 1

27ν2)x) ,

j2P3

(x ; ν) =1 +

(3 + 1

3ν − w6(ν)

)x +

(9− 17

4ν + 1

9ν2 −

(3 + 1

3ν)w6(ν)

)x2

x(1− w6(ν)x

) ,

where

w6(ν) ≡ 192

(36− ν)(9− 4ν)w5(ν).

The test-mass limit is recovered exactly,

limν→0

j2P1

(x ; ν) = limν→0

j2P2

(x ; ν) = limν→0

j2P3

(x ; ν) =1

x(1− 3x).





The EOB metric coefficient −g eff00 (u) = A(u) (with u ≡ 1/q′)

has the following 3PN-accurate truncated Taylor expansion:

A(u; ν) = 1− 2 u + 2ν u3 + a4(ν) u4 +O(u5),

where the 3PN coefficient a4(ν) reads

a4(ν) =

(94

3− 41

32π2

)ν.

We want now to factor a zero of A(u; ν) (and no longer a pole),which will be smoothly connected with the test-mass-limit (i.e.ν → 0) value u = 1/2, therefore the Pade-improved APn(u; ν) ofA(u; ν) we define at the nPN level as

APn(u; ν) ≡ P1n [Tn+1[A(u; ν)]] .













EOB formalism ←→ synergy ←→ Numerical Relativity

Select sample of NR results↓

Calibrating EOB flexibility parameters↓

NR-improved EOB(accurate ‘analytical’ waveform templates; ‘analytical’ predictions)





Flexibility parameters in the metric coefficient −g eff00 (u) = A(u)

Instead of using (maybe in Pade-improved form)

A3PN(u; ν) = 1− 2 u + 2ν u3 + a4(ν) u4,

one considers two-parameter class of extensions of A3PN(u; ν)defined by

A(u; ν, a5, a6) ≡ P15

[A3PN(u; ν) + ν a5 u

5 + ν a6 u6].





“Contrary to the hint of Mroue, Kidder, and Teukolsky (2008), thatEOB is ‘only . . . a very good fitting model’, we think that EOBincorporates a lot of the real physics of coalescing BBH, and thatits parametric flexibility is a way to complete it with the ‘missing’non-perturbative physics provided by NR simulations.”

(T.Damour, 2008)


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